Australia had set the hosts - who chased down 202 to win the Twenty20 international in Rajkot on Thursday - a total of 305, spearheaded by Bailey who chalked up 10 boundaries off 82 deliveries.
India opener Shikhar Dhawan went cheaply when he feathered Faulkner to wicketkeeper Brad Haddin for seven - although Haddin then had to the leave the field after being poked in the eye by the bowler's attempted high-five.
Rohit Sharma's resistance was ended for 47, including six fours, when he edged Shane Watson through to stand-in glovesman Phil Hughes and India were 66 for two.
Faulkner then delivered what proved to be the key wicket of Suresh Raina, out for 39, with the total fast-tracked up to 137 as India threatened to match the required seven-over run-rate, before Yuvraj Singh went for seven.
When Virat Kohli was trapped leg before by Watson for 61, coming off 85 balls with four boundaries, India's middle order could not produce a response.
Captain Mahendra Singh Dhoni managed a quick 19 from his 22 deliveries, but when he was skittled out by Clint McKay, the run chase was as good as over at 197 for seven off 41 overs.
Vinay Kumar battled on to 18, but Australia soon closed out a 72-run victory inside the distance.
Faulkner finished with three for 47 from his eight overs, while McKay took two for 36 off 10 and Watson two for 31 from eight.
The two sides meet again at the Sawai Mansingh Stadium in Jaipur on Wednesday for the second match of a seven-game series.
Earlier, Bailey had chosen to bat and following a slow start by the tourists which saw them creep to 13 from the first six overs, Aaron Finch and Hughes, coming in at the top of the order, eventually forged a decent opening partnership as they put on 110.
Hughes fell three short of a half-century when he was caught by Suresh Raina off Ravindra Jadeja, and Yuvraj Singh then accounted for Shane Watson who lasted only four balls, scoring two.
However Bailey, with the help of 72 from Finch - playing at his home Indian Premier League venue - helped push Australia on to 146 before the opener went, again off the bowling of Yuvraj Singh, at the start of the 28th over.
Having arrived when his side had just slipped to 113 for two, Bailey's fine innings saw the tourists move through 250 before he was eventually dismissed when caught by Raina off Ravi Ashwin.
Glenn Maxwell, another change to the side, added 31 and Faulkner 27 as Australia batted out their full 50 overs, with Mitchell Johnson and McKay unbeaten for nine and 11 repsectively.
Yuvraj Singh was the pick of the India bowlers, with two wickets - his first in one-day competition since the 2011 World Cup - for 34 from six overs.